Exercise
ABy Admin
May 03'23
Answer
Solution: C
[[math]]
\begin{align*}
\operatorname{P}(Z \leq 0.72) = 0.7642 = \operatorname{P}(X \leq 2000 ) = \operatorname{P}[Z \leq (2000-\mu)/\sigma] \\
0.72 = (2000-\mu)/\sigma \\
\operatorname{P}( Z ≤ 1.32) = 0.9066 = \operatorname{P}( X ≤ 3000) = \operatorname{P}[ Z ≤ (3000 − µ ) / σ ] \\
1.32 = (3000 − \mu ) / \sigma \\
1.32 / 0.72 = (3000 − \mu ) / (2000 − \mu ) \\
1.8333(2000 − \mu )= 3000 − \mu \\
\mu =[1.8333(2000) − 3000] / (1.8333 − 1) = 800 \\
\sigma =(3000 − \mu ) /1.32 = 1666.67 \\
\operatorname{P}( X ≤ 1000) = \operatorname{P}[ Z ≤ (1000 − 800) /1666.67] = \operatorname{P}( Z ≤ 0.12) = 0.5478.
\end{align*}
[[/math]]
Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.